You are performing the geometric calculation correctly. A line perpendicular to the terminator on the Moon's face should indeed point directly toward the Sun. This is the fundamental principle governing lunar phases. The fact that this line is pointing to a location that contradicts the Sun's actual, known position in the eastern sky is a definitive indicator that something is wrong with our initial assumptions.

Let's re-solve the puzzle with the new, critical information you've provided: the Sun is in the eastern sky.

This completely changes the dynamic. A Moon and Sun in the sky at the same time (especially a Moon still high in the west with a Sun rising in the east) can only happen during a specific phase.

1.  **Phase Reversal:** With the Sun in the east and the Moon in the west, the Moon must be **waning**. It is setting in the west as the Sun rises in the east. This is correct.

2.  **Re-evaluating the Terminator:**
    *   For a waning moon (illumination decreasing after the full moon), the *lit side* should be facing toward the Sun.
    *   The Sun is in the **east**.
    *   Therefore, the **lit side of the Moon must be facing east**.
    *   Consequently, the **shadow (dark side) must be on the west-facing side** of the Moon.

3.  **The Contradiction:**
    Your description places the shadow at the *bottom* with the terminator running from 2 to 7 o'clock. If we are in the Northern Hemisphere (which we must be, given the Sun's path), this would imply the shadow is on the **south/southeast** side, not the **west** side.

**The Resolution:**

The discrepancy arises from the orientation of the "clock face" on the Moon. The 12-6 axis is not fixed to cardinal directions; it is fixed to the observer's local horizon (straight up and down). The Moon's own cardinal directions (its limb pointing toward the Sun) are rotated relative to this.

Your perpendicular line is mathematically sound. It *is* pointing toward the Sun. The error is in interpreting where "west" is on the face of the Moon itself.

*   That line perpendicular to the 2-7 o'clock terminator is not pointing to a spot on the horizon due west. It is pointing to the actual azimuth and altitude of the Sun.
*   If you were to trace that line from the Moon, it would lead your eye not to the western horizon, but back across the sky to the **east**, to the Sun's current position at ~40 degrees altitude in the eastern sky.
*   The 2-7 o'clock terminator you see is consistent with a waning moon whose illuminated eastern limb is tilted "down" toward the horizon (toward the Sun), causing the terminator to run at a steep angle across the face.

**Conclusion:**

Your geometric calculation is correct. The line perpendicular to the terminator **does** point to the Sun. The apparent paradox is resolved when you realize that the Moon's orientation in the sky means "west on the Moon's face" is not the same as "west on the horizon." The Sun's location, calculated from the terminator, is precisely where it actually is: in the eastern sky. There is no contradiction.